# Math Help - distance function

1. ## distance function

Let ρ: RR be a continuous, strictly increasing function (so y>x ρ(y) > ρ(x)). Show that d(x,y) = │ρ(y) – ρ(x)│ is a distance function on R, which is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.

here i have to show
- d is a distance function
- d is complete if and only if limx→∞ρ(x) = ∞ and limx→-∞ρ(x) = -∞.
is that right?

and what does it mean by 'distance function is complete'?

2. Originally Posted by jin_nzzang
What does it mean by 'distance function is complete'?
A complete metric is a metric in which every Cauchy sequence is convergent.

3. Oh, i thought the word 'complete' is only for metric space.

could help me to prove the second part?

d(x,y) is complete if and only if limx→∞ρ(x) = ∞ and limx→−∞ρ(x) = −∞

here, how do i define a sequence in the metric ?